Defender Osaze Urhoghide signed a four-year deal on Thursday afternoon.
Postecoglou revealed the 20-year-old is a player the club have been tracking for a while.
The latest name to be linked with a switch to Celtic is Australian international Aaron Mooy, who would very much be a Postecoglou signing.
According to the Daily Record, the 30-year-old midfielder is a target but it would mean the player taking a pay cut. He currently plays in China for Shanghai Port where he is a team-mate of former Chelsea star Oscar.
Mooy is very familiar to Postecoglou having won the majority of his 43 caps for Australia under the management of the new Celtic boss.
The player may be familiar to Scottish football fans having played 30 times for St Mirren at the start of his career.
He departed the Buddies and returned to Australia where, across four seasons, first with Western Sydney Wanderers then Melbourne City, progressed to become an influential midfielder.
Due to Melbourne’s links with Manchester City he made the move to England before being loaned out to Huddersfield Town.
Lines and space
His ability and vision on the ball was key in a deep midfield role, labelled the “architect” as the Terriers won promotion to the Premier League. The loan was turned into a permanent deal for £8million, rising to £10million.
Mooy moved on to Brighton and became a key component for manager Graham Potter as he changed the style and identity of the team's play.
“I think he fits in well to the combination of players we have,” Potter said. “He likes to play behind the lines and in space. Some players find that difficult – they want to get on the ball or have touches of the ball.
“Aaron sees the game well enough to be able to understand when he is in space and when he is in good space, he can stay in it and wait for the ball to come.”
Mooy is a willing runner who will happily do the dirty work in midfield. But he's at his best in possession, progressing play and getting the ball into the final third.
After nearly a decade, he could be set for a Scottish return.